In this paper we study the complexity of some size constrained clustering problems with norm Lp . We obtain the following results: (i) A separation property for the constrained 2-clustering problem. This implies that the optimal solutions in the 1-di

Gene selection methods aim at determining biologically relevant subsets of genes in DNA microarray experiments. However, their assessment and validation represent a major difficulty since the subset of biologically relevant genes is usually unknown. To solve this problem a novel procedure for generating biologically plausible synthetic gene expression data is proposed. It is based on a proper mathematical model representing gene expression signatures and expression profiles through Boolean threshold functions. The results show that the proposed procedure can be successfully adopted to analyze the quality of statistical and machine learning-based gene selection algorithms.

In this paper we consider the classes REC1 and UREC1 of unary picture languages that are tiling recognizable and unambiguously tiling recognizable, respectively. By representing unary pictures by quasi-unary strings we characterize REC1 (resp. UREC1) as the class of quasi-unary languages recognized by nondeterministic (resp. unambiguous) linearly space-bounded one-tape Turing machines with constraint on the number of head reversals. We apply such a characterization in two directions. First we prove that the binary string languages encoding tiling recognizable unary square languages lies between NTIME(2n) and NTIME(4n); by separation results, this implies there exists a non-tiling recognizable unary square language whose binary representation is a language in NTIME(4n log n). In the other direction, by means of results on picture languages, we are able to compare the power of deterministic, unambiguous and nondeterministic one-tape Turing machines that are linearly space-bounded and have constraint on the number of head reversals.

BACKGROUND: The unsupervised discovery of structures (i.e. clusterings) underlying data is a central issue in several branches of bioinformatics. Methods based on the concept of stability have been recently proposed to assess the reliability of a clustering procedure and to estimate the "optimal" number of clusters in bio-molecular data. A major problem with stability-based methods is the detection of multi-level structures (e.g. hierarchical functional classes of genes), and the assessment of their statistical significance. In this context, a chi-square based statistical test of hypothesis has been proposed; however, to assure the correctness of this technique some assumptions about the distribution of the data are needed. RESULTS: To assess the statistical significance and to discover multi-level structures in bio-molecular data, a new method based on Bernstein's inequality is proposed. This approach makes no assumptions about the distribution of the data, thus assuring a reliable application to a large range of bioinformatics problems. Results with synthetic and DNA microarray data show the effectiveness of the proposed method. CONCLUSIONS: The Bernstein test, due to its loose assumptions, is more sensitive than the chi-square test to the detection of multiple structures simultaneously present in the data. Nevertheless it is less selective, that is subject to more false positives, but adding independence assumptions, a more selective variant of the Bernstein inequality-based test is also presented. The proposed methods can be applied to discover multiple structures and to assess their significance in different types of bio-molecular data.

BACKGROUND: Cluster analysis has been widely applied for investigating structure in bio-molecular data. A drawback of most clustering algorithms is that they cannot automatically detect the "natural" number of clusters underlying the data, and in many cases we have no enough "a priori" biological knowledge to evaluate both the number of clusters as well as their validity. Recently several methods based on the concept of stability have been proposed to estimate the "optimal" number of clusters, but despite their successful application to the analysis of complex bio-molecular data, the assessment of the statistical significance of the discovered clustering solutions and the detection of multiple structures simultaneously present in high-dimensional bio-molecular data are still major problems. RESULTS: We propose a stability method based on randomized maps that exploits the high-dimensionality and relatively low cardinality that characterize bio-molecular data, by selecting subsets of randomized linear combinations of the input variables, and by using stability indices based on the overall distribution of similarity measures between multiple pairs of clusterings performed on the randomly projected data. A chi2-based statistical test is proposed to assess the significance of the clustering solutions and to detect significant and if possible multi-level structures simultaneously present in the data (e.g. hierarchical structures). CONCLUSION: The experimental results show that our model order selection methods are competitive with other state-of-the-art stability based algorithms and are able to detect multiple levels of structure underlying both synthetic and gene expression data.

Clustering analysis of gene expression is characterized by the very high dimensionality and low cardinality of the data, and two important related topics are the validation and the estimate of the number of the obtained clusters. In this paper we focus on the estimate of the stability of the clusters. Our approach to this problem is based on random projections obeying the Johnson-Lindenstrauss lemma, by which gene expression data may be projected into randomly selected low dimensional suhspaces, approximately preserving pairwise distances between examples. We experiment with different types of random projections, comparing empirical and theoretical distortions induced by randomized embeddings between Euclidean metric spaces, and we present cluster-stability measures that may be used to validate and to quantitatively assess the reliability of the clusters obtained by a large class of clustering algorithms. Experimental results with high dimensional synthetic and DNA microarray data show the effectiveness of the proposed approach.

In this paper, a genetic model based on the operations of recombination and mutation is studied and applied to combinatorial optimization problems. Results are: 1. The equations of the deterministic dynamics in the thermodynamic limit (infinite populations) are derived and, for a sufficiently small mutation rate, the attractors are characterized; 2. A general approximation algorithm for combinatorial optimization problems is designed. The algorithm is applied to the Max Ek-Sat problem, and the quality of the solution is analyzed. It is proved to be optimal for k > or = 3 with respect to the worst case analysis; for Max E3-Sat the average case performances are experimentally compared with other optimization techniques.

In this work we study the relationship between PAC learning and the property of uniform convergence. We define the concept of polynomial uniform convergence of relative frequencies to probabilities in the distribution–dependent context. Let Xn = (0,1)n, let Pn be a probability distribution on Xn and let Fn⊂2xn be a class of events. The family {(Xn, Pn, Fn)}n≥1 is said to be polynomially uniformly convergent if, for all n, the probability that the maximum difference (over Fn) between the relative frequency and probability of an event exceed a given positive ε is at most δ (0 < δ < 1), when the sample on which the frequency is evaluated has size polynomial in n, 1/ε, 1/δ. Given at-sample(x1,…,xt), let Cn(t)(x1,…,xt) be the Vapnik-Chervonenkis dimension (VCdim) of the set (x1,…xt ∩ f | f ϵ Fn and M(n,t) the expectation E(Cn(t)/t). The results we obtain are:

Enumeration problems constitute a major part of combinatorial mathematics.